Solving Systems of Linear Equations and Matrices

Show that for this system to be consistent, the constants a, b, and c must satisfy c = a + b.

2. Show that the elementary row operations do not affect the solution set of a linear system.

3. Consider the system of equations

ax + by = k
cx + dy = L
ex + fy = m

If the system of equations is consistent, explain why at least one equation can be discarded from the system without altering the solution set.

4. Consider the system of equations

ax + by = k
cx + dy = L
ex + fy = m

If k = L = m = 0, explain why the system must be consistent. What can be said about the point of intersection of the three lines if the system has exactly one solution?

5. We could also define elementary column operations in analogy with the elementary row operations. What can you say about the effect of elementary column operations on the solution set of a linear system? How would you interpret the effects of elementary column operations?

Solution Preview

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1. Consider the system of equations

x + y + 2z = a
x + z = b
2x + y + 3z = c

Show that for this system to be consistent, the constants a, b, and c must satisfy c = a + b.